Pressure drop and mass transfer around perforated turbulence promoters placed in a circular tube

I,,,. J. Hrur Muss Trunsfi~. Pnnted I” Great Britam

0017-9310/91$3.00+0.00 ,I‘ 1991 Pergamon Press plc

Vol. 14, No. 8, pp. 1909-1916, 1991

Pressure drop and mass transfer around perforated turbulence promoters placed in a circular tube HWA

WON

RYU,? HO

t Department $ Department

NAM

YOUNG

SOON

CHANGt$

HYEONJ

DONG

IL

LEE,3

and 0 OK PARK?

of Chemical Engineering, Korea Advanced Institute of Science and Technology, P.O. Box 13 1, Cheongryang, Seoul 130-650, Korea of Chemical Engineering, Chonnam National University, Kwangju 500-757, Korea (Rccriwd

14 Septemhrr

1989 und in,final,form

22 July 1990)

Abstract-The friction factors and mass transfer coefficients in a tube with fins of e/D = 0.1,0.2,0.24 and 0.3 are investigated both experimentally and theoretically in the wide range of Reynolds numbers up to 8000. As expected the friction factor decreased linearly with Reynolds number up to Re = 200 and remained nearly constant at Re > 500 for all cases of c/D. The distribution of pressure between the fins is measured in detail. The perforated fins are proved to be effective both in reducing loss and in enhancing mass transfer.

1. INTRODUCTION THE ENHANCEMENT of

the heat or mass transfer from the flowing fluid in a channel is important for the design of compact heat exchangers, electrodialysers, and membrane blood oxygenators [I]. Turbulence promoters have been widely used for this purpose. In electrodialysers, for example, the turbulence promoters installed in a channel prevent the formation of the boundary layers and thus they not only improve the mass transfer but also prevent polarization. Most previous investigations were concerned with turbulent flow at high Reynolds number, but it is much more necessary to promote the heat or mass transfer augmentation in the case of laminar flow because this case usually results in low heat or mass transfer. Many types of turbulence promoters have been proposed to see the global mass transfer enhancements, but few studies have been carried out on the distribution of local mass transfer coefficients in the laminar flow with fin-type turbulence promoters. Rowley and Patankar [2] studied the heat transfer in a tube with fins riumerically for laminar flow. They showed that although the recirculating flow aids mixing, the heat transfer coefficient often decreases in the laminar flow due to the poor heat exchange between the main stream and the vortices near ribs. Thomas [3] and Miyashita et al. [4] used detached turbulence promoters with some clearance between the wall of the channel, and found that the mass transfer increases throughout the channel including the region just after the turbulence promoters, by comparing with the mass transfer in a smooth channel. The perforated fins can be introduced as a compromise between these two types of promoters. In fact, Tanasawa et al. [5] used perforated g Author

to whom all correspondence

should be addressed.

plate promoters to see an increase of heat transfer coefficients at high Reynolds number. In addition, they showed that perforated promoters have an advantage of low pressure drop since the form drag due to the installed promoters can be reduced significantly. Therefore, it may be interesting to perform the mass transfer experiments on flows of low Reynolds number in a tube with perforated and non-perforated fins. In addition, the pressure measurements were carried out to determine the friction loss due to the installation of the fins. Furthermore, theoretical solutions were obtained using a finite difference method in the laminar flow regime and were compared with experimental results.

2. EXPERIMENTAL

APPARATUS

AND

PROCEDURES 2.1. Pressure

measurement

Figure 1 shows a schematic diagram of the experimental apparatus for pressure measurements. Water

Fw. I. Schematic diagram of the experimental apparatus : B. bottom tank ; D, smooth downstream : F, feed ; H. head tank ; 0, over flow ; P. pump; R, rotameter; T, test section for local pressure drop; U, smooth upstream section; V. control valve. 1909

NOMENCLATURE A

c C, Cl,, c;,

C, n

Q 1’ f

F

electrode area [m’] concentration divided by C,,. [--I bulk concentration [kmol n-‘1 inlet concentration [kmol rn- ‘J surface pressure coefficient [-] concentration at a solid wall [kmol m- ‘1 inside diameter of a tube [m] diffusivity [mZ s ‘1 height of roughness element [m] friction factor I--j Faraday constant. 96 520 C k mol- ’ limiting current density [A m ‘1 local mass transfer coefficient [m s ‘1 pipe length, L’D [-] valence charge of ion I-] mass flux (kmol m ’ s ‘1 dimensionless pressure, p’/pU’ [-] local static pressure [m’] static pressure at reference tap [N m ‘1 Peclet number, CID/D, [--1

at 25 C was supplied from the constant head tank (H) and passed through the flow control valve (V). rotameter (R) and test section (T). The test section was 2676 mm long and 17 fins were inserted in it equidistantly. The fin-inserted section was preceded by a smooth tube (U) 650 mm long and WdS followed by a tube (D) of the same length. Total pressure loss was calculated by measuring the pressures just before the first fin and just after the last fin. The local pressure distribution between the ninth and tenth fins was measured using differential manometers that were connected to the 12 tabs as shown in Fig. 2. Carbon tetrachioride or mercury was used as the manometer fluid depending upon the magnitude of the pressure differences. Manometer readings were carefully measured with a cathetometer (Gaertnet Scientific Corp., U.S.A.) within the accuracy of 0.01 mm. The inner

I RP Sl7 s/7,,,

Lf 3

radial coordinate [ -1 Reynolds number based on L) and Cf[ -f Sherwood number, kDiD, [-_I mean Sherwood number [--I average velocity based on tube diameter [ms ‘I axial coordinate [-_I.

Greek symbols fraction of the perforated area in a P tin [-_I density of fluid [kg m ‘1 I’ diameter of drilling holes on the ribs 4 [mm] dependent variables Yr~/r, and C I-1 stream function, ul’j(DU) I---) Y vorticity, n/D/U [--I. 0) Superscript variable with dimensions

diameter of the tube was 49.4 mm and the orifice diameters were 19.4,25.4,29.4, and 39.4 mm, thus the ratios of the fin height to tube diameter (e/n) were 0.3, 0.24, 0.2, and 0.1, respectively. Both tube and fins were made of acryl and the thickness of the rib was 2.0 mm. The pitch of the fins was fixed at 87 mm. Open area ratios (/J), defined by the ratio of the total perforated area to the fin area, of the perforated fins of e/D = 0.3 were made 0.08, 0.13, and 0.23, respectively, by drilling holes of 1.6-2.1 mm in diameter on the ribs. Two types of the hole array were madestaggered and in-line type. Detailed configurations of the perforated fins are shown in Fig. 3. 2.2. E.uperirnent on muss trunsj~~r The mass transfer experiment was performed by the limiting current method [6] by utilizing the well-

FIG. 2. Details of the test section for local pressure

distribution

(unit mm).

Pressure

drop and mass transfer

around

perforated

STAGGERED

IN-LINE

/ ; \

6

b-rn43.4

\

n=O.O8

\

r3= 0.13

b=

1.7(96)

(0y&ip2.5 \\ .__M

I

/I

I-I= 0.23 6=1.7(64). 2.1(64)

0;

--O-= ‘b. 23 1.7(64). 2.1(64)

FIG. 3. Configurations of the perforated fins (numbers of the holes are given in parentheses, e/D = 0.3, unit mm).

(1)

Sh = kD/D, = iDi(nFC,D,).

3. THEORETICAL

/

‘__*’ @= 1.7(96)

1911

tube

The dimensionless mass transfer rate and the Sherwood number can be obtained as

6= 1.6(54)

: \

placed in a circular

of 600 mV was supplied using a d.c. power supply. At the limiting current the interfacial concentration, C,, becomes zero, and the mass transfer coefficient is expressed as

-q 6= 1.6(G4)

I

n= 0.13

promoters

k = i/(nFC, -C,)). 9

n= 0.08

turbulence

(2)

CALCULATION

A theoretical study was carried out for the typical module shown in Fig. 5 by a two-dimensional, finitedifference method in a cylindrical coordinate. The flow was assumed to be laminar. It is a somewhat different story to treat a turbulent flow problem with flow separation, which is beyond the scope of this paper. Computations were performed for different values of the Reynolds number and of the geometrical parameter e/D. The flow patterns were assumed to be periodical, while the profiles of concentration were not because mass transfer was assumed to occur at only one section among the repeated ribs. The governing equations for the stream function, vorticity and concentration can be written as follows :

(3) known redox reaction between ferricyanides and ferrocyanides. The electrolytic solution consisted of 0.01 M of K,Fe(CN),, 0.01 M of K,Fe(CN), and 1.0 M of NaOH. The concentration of the ferricyanide was checked periodically before each experiment by using the iodometric method [7]. To prevent decomposition of the ferricyanide all experiments were carried out in a dark room with local illumination and nitrogen was continuously bubbled through the fluid at the bottom tank (B in Fig. 1). The experimental apparatus for mass transfer is not presented here but it is similar to the apparatus shown in Fig. 1 with some minor modifications. Equipment for measuring the currents consisted of a d.c. power supply, two multimeters and a control box. Nickel plates 0.1 mm thick were attached to the acryl tube by epoxy resin. Sixteen cathodes 4.72 mm wide were installed and circular teflon sheets 0.45 mm thick were inserted between them for electrical isolation. Great care was paid to ensure smooth conditions at tube walls. Detailed locations of the cathodes are presented in Fig. 4. The test section for mass transfer was located at the same test section for pressure measurement as explained before. The anodes 600 mm in length were installed downstream of the cathodes. The physical properties of the solution used in this experiment and the details of the electric circuit can be found elsewhere [8]. Typical polarization curves were obtained for various Re up to 500 and limiting currents were obtained at above 400 mV. A constant voltage

dC

X

I

“‘.-g+“qg=p,

av

i ac

(

aY’+;pL:z’

The stream function and the vorticity equations are defined as follows :

i ay v,.=

The boundary

-p-,

r oz

conditions

L’, =

c?v

>

(5)

in the above

1 l?P -7

(6)

r CY

for stream function

Y are :

(i) at all tube walls y’=0, (ii) the relations

%O; an

(8)

between the inlet and the outlet y,,(r)

= Y,,,(r)

Yr, (r) = YIW, (r) ; (iii) at the centre of the tube

(9)

(10)

1912

H. W. Ruu CI ul Unit:mm -I

-87

r

Teflon

Teflon

FIG. 4.Details of the test section for local mass transfer (unit mm).

H ~ = 0.

(11)

dr

The boundary

conditions

for concentration

These three governing equations formed into the following generalized

can be transform :

C are :

(i) at the inlet c=

I;

6C -=

0.

(12)

(ii) at the outlet

a:

(iii) at the conducting of limiting current

(13)

’

tube wall under the condition

c=o; (iv) at the non-conducting

(14) walls

ac an

(1%

-=O

where n stands for the normal distance

from the wall.

2 --___----__-----__--_o

--I-

OFLOW

Therefore, the pressure drop along the z-axis can be calculated by integrating the pressure gradient between the neighbouring points as

OlRECTlON

-

JW

0.5 JN

II

CONOUCTING WALL

( 17)

J,

P

0 0.04 I1 IWl

where 4 denotes the dependent variables Y, w/r, and C. The corresponding coefficients a+, b,, cd, and d,, are listed in Table 1. The upwind difference method developed by Gosman et al. [9] was also used to solve the above equations. The linear variation of vorticity from the wall to the neighbouring point was assumed. The mesh size ranged from I/ 100 near the wall to 1/SO near the centre of the tube in the r-direction and from h = l/50 to l/l0 in the z-direction. The underrelaxation method was used to give a stable solution at the expense of long computation time. From the Navier-Stokes equation the pressure gradient along the z-axis can be obtained as follows :

II 1.7 1.74 IW2 IN

FE. 5. Model for theoretical analysis of the tube with circular fins (lengths are scaled by inside diameter of the tube).

Table 1. Coefficients of equation (I 6)

Y

0

co/r

r2

c

1

I/r’

Y2 1

1

-w/r

1/Re

0

l/(ReSc)

0

Pressure

R -dP

drop and mass transfer

s(> A

as

ds=p,-p,2;

K

g

11

around

perforated

(sFl-s*)

(18) where s is the distance along an integration path. Friction loss was evaluated by the following equation :

f’= -AP’/(4L’/D)/(PU’/2) The rate of mass transfer N= In dimensionless

-D,

= -Ap/(2L).

(19)

at the tube wall is

ac

0 dr

=k(C;,-C;).

(20)

form

. The mean Sherwood

number

Sh, = ;

(21)

was defined as

L Sh dz. s0

(22)

turbulence

promoters

1913

placed in a circular tube

constant e/D (=0.3) and the results plotted in Fig. 7. One can easily find that the decreasing pattern of friction factor is similar to the non-perforated ones, however, the perforated fins have lower friction factors. The friction factor at /3 = 0.23 does not seem to reach a constant value even for the large Reynolds number region. Though the friction factors are still larger than those of smooth tubes they significantly decrease as p increases in the high Reynolds number range. One can see that tubes with staggered perforated fins have slightly higher friction factors than those with in-line perforated fins at Reynolds numbers higher than 50. For the qualitative explanation on the differences in friction factors it is necessary to investigate the local pressure distributions between ribs, which can be either non-perforated or perforated, in detail. The local pressure distribution in a section between the two fins can be represented by the surface pressure coefficient (C,) at the ith tap which is defined as follows :

c/l = (P:-P6)l(Pu2/2)

(23)

where pi is the static pressure at the reference tap (first

4. RESULTS AND DISCUSSION 4.1. Pressure drop Figure 6 shows the relationship between friction factor and Reynolds number in a tube with non-perforated fins with e/D = 0.10. The friction factor decreases with the average slope of - 1.0 until Re = 200 and is almost constant at Re > 500. In the cases of e/D = 0.2 and 0.3 the friction factors show the same trends with Reynolds number. The slope of the theoretical solutions agrees well with the experiments until about Re = 200, but disagrees at larger Reynolds numbers. The reason is that the actual flow pattern becomes unstable at Reynolds numbers above 200, while the theoretical solution is obtained stable due to the assumption of the laminar flow. The experiments on the effect of perforation were carried out for several kinds of perforated fins with

tap as shown in Fig. 2), pi the ith local static pressure, and U the average velocity based on the diameter of a tube. The dependence of C,, on e/D at Re = 200, /3 = 0 is found in Fig. 8. When e/D = 0.1, 0.2, and 0.24, C, shows a little scattered pattern, but it is quite gradual. A big difference can be found in the local pressure distribution for e/D = 0.3 where C,, tends to have a minimum point near the centre because of the recirculating flows generated between the ribs, as demonstrated in Fig. 9. As Chang [lo] pointed out

100 e/D = 0.30

0

10 -

100 -

e/o = 0.10 Y-

0 AH : 0 A Cl : 8

O.‘t

0.01 1

I 10

I 100

I 1000

I 10000

10000c

Re FIG. 6. Friction factor for a tube with non-perforated (e/D = 0.I).

fins

IN-LINE

STAGGERED

: NON-PERFORATED

\

1000

Re FIG. 7. Friction factor for a tube with perforated

fins

1914

H. W.

RYU

('1 cd.

1 e/D

0

0.10 0.20 -2

0.24

-4 Re = 8000 NONPERFORATED -6

-8 0"

e/D

0:

0.10

A:

0.20

0:

0.24

0:

0.30

DIMENSIONLESS FIG.

-iO

DISTANCE

FROM

UPSTREAM

$

FIN

8.

Distribution of the surface pressure coefficient at Rc = 200 for the case of non-perforated fins as functions of dimensionless distance from upstream fin scaled by inside diameter of the tube.

0

0,4

DIMENSIONLESS

088 DISTANCE

1.2 FROM

0.30

1,6 UPSTREAM

FIN

FIG.

before, the low pressure near the centre of the bottom and the high pressure at the corner are typical of a single, stable vortex generated. The pressure recovers gradually at the last tap near the neighbouring fin even though the pressure recovery is not good at high c/D as shown in Fig. 8. On the other hand at a high Reynolds number of 8000, the pressure between nonperforated fins decreases very slowly and linearly for all e;D as is presented in Fig. IO, and one can easily find that the pressure is not recovered at all at the last tap. This kind of stable pressure distribution results in the almost constant friction factor at high Reynolds number. Figure I I shows the effect of the open area ratio, 8, on pressure distribution at a low Reynolds number of 200. The distribution of the local pressure in tubes with perforated fins shows a great difference from those with non-perforated fins indicating that a recirculating vortex disappears by the perforated holes. From a dye-injection experiment it was con-

012

FIG. 9.

0:4 Contours

0:6 ofstream

0.8

10.Distribution of the surface pressure coefficient at Re = 8000 for the case of non-perforated fins as functions of dimensionless distance from upstream fin scaled by inside diameter of the tube.

firmed that some portion of the fluid passed through the holes perforated on the ribs, and thus the vortex disappeared resulting in an unstable flow pattern. Figure I2 also shows the distribution of the local pressure in a tube with perforated fins at a high Reynolds number of Re = 8000 and e/D = 0.3. The local pressure distribution for several fl shows a similar tendency to the non-perforated case except the amount of the permanent pressure losses.

Figure I3 shows a comparison of the distribution of the local mass transfer coefficients at various B and at nearly the same Reynolds numbers. Comparing with the data of the non-perforated fins. one can find

1.0 function

(r/D

1:2 = 0.3,

114

Re = 200).

1.6

1.74

Pressure

drop and mass transfer

around

perforated

turbulence

promoters

placed in a circular

1915

tube

180

1

160

140

6iN-LINE 0.23 A: IN-LINE 0.13

q:IN-LINE

0.08 80

60

0

0,4

D~~~ENs~oNLESS

0.8 DISTANCE

1.2 FROM

1.6 UPSTREAM

FIN

FIG. Il. Distribution of the surface pressure coefficient at Re = 200 and e/D = 0.3 for the case of perforated fins as functions of dimensionless distance from upstream fin scaled by inside diameter of the tube.

that mass transfer is increased significantly by perforation. The reason for this is that the fluids that pass through the the perforated holes break both the recirculating flow and the concentration boundary

-

Q 0.23

0

C.4

DIMENSIONLESS FIG.

12.

0.8 DISTANCE

1.2 FROM

1.6 UPSTREAM

FIN

Distribution of the surface pressure coefficient at Re = 8000 and e/D = 0.3 for the case of perforated fins as functions of dimensionless distance from upstream fin scaled by inside diameter of the tube.

40

FIG.

13. Dependence of the local Sherwood number on open area ratio as functions of dimensionless distance from upstream fin scaled by inside diameter of the tube.

layer near the tube wall. The mean Sherwood numbers obtained by both experiment and theory are shown in Fig. 14 for non-perforated cases (e/D = 0.1). Sherwood number increases with Re” with a = 0.37 until a moderate Reynolds number and thereafter approaches a constant value. This tendency is also found for larger e/D at a slightly larger value of a (a = 0.43 when e/D = 0.3). Figure 15 shows the enhancement of the mean Sherwood numbers by perforation at e/D = 0.3. For comparison the mean Sherwood number for the inlet section of the same length as the fin pitch usedin this experiment is also presented as a dotted line in the figure [ 1I]. One can see that the perforated fins are superior to both the non-perforated fins and the entrance section of a smooth tube in mass transfer. The non-perforated fins, however. are still effective for promotion of mass transfer com-

L--+-J

0.01

0.1

10

100

1000

Re

FIG. 14. Comparison of the theoretical values with experimental data for mean Sherwood number.

the mass transfer, however, the in-line type is thought to be a little better than the staggered one. AckrzoM,/eclymlents--Two of the authors (H. W. Kyu ~11 D. I. Lee) gratefully acknowledge the financial support of ihissworkby the Korea Scienceand Engineering Foundation. REFERENCES

L,t’ ,,,’

i

/

I

/

,liO

213

I

i:o

43

He

Frc;. 15. Dependence

of the mean Sherwood type.

number

on fin

pared with the lower limit of the Graetz problem which the limiting value of S1z is 3.24 [I I].

of

5. CONCLUSIONS

Friction factors and mass transfer coefficients in a tube with fins of e/o = 0.1, 0.2, 0.24. and 0.3 were investigated in the Reynolds number range up to 8000. Comparing the results of the theory and experiment we found that the flow becomes unstable above Re = 200. The local distribution of the pressure between the neighbouring fins was investigated in detail and it was found that the flow through the holes in the perforated fins breaks the recirculating vortices between the fins and thus the friction loss is reduced, and also it reduces the concentration boundary layer near the tube wall so that the mass transfer is enhanced. The detailed configuration of perforation does not have much effect on both friction factor and

I. Ho Nam Chang and Joong Ken Park. Hm/ho~~k of Hcut rmd MUSS Trctn.s/& 0per~~tion.s (Edited by N. P. Cheremisnoff), Vol. II. Gulf, Houston, Texas (1986). 2. G. J. Rowley and S. V. Patankar. Analysis of laminar Ilow heat transfer in tubes with internal circtin~fcrential fins. ht. J. Hmr Mm Trrrrr~~~c~ 27, 553 560 (1984). 3. D. G. Thomas. Forced convection mass transfer, Part III. Increased mass transfer from a Ilat plate caused by the wake from cylinders located near the edge of the boundary layer, A.I.Ch. E. JI 12, 124--l 30 ( 1966). Kf&rr 4. H. Miyashita, Y. Shiomi and K. Wakabayashi, &r&u ~(~~~JZ~~.S~Z~~ 7, 349 -354 ( 198 I )_ S. Nishio. K. Takano and M. Toda. 5. I. Tanasawii, Enhancement of forced-convection heat transfer in a rectangular channel using turbulence promoters, J. S’oc. Me&. Engn~g Jupun (in Japanese) 49B, 676 (1983). 6. H. N. Chang, H. W. Ryu, D. II. Park. Y. S. Park and J. K. Park. Effect of external laminar channel flow on mass transfer in a cavity. ltr!. 1. Ncur Mas ‘Trim,+7 30, 2137 2139 (1987). R. Belcher. Chi~nreiric .4~~~/~,.s~.~. Vol. 7. I. M. Kolthoff‘and III. Interscience, New York (1957). 8. D. H. Kim, I. H. Kim and H. N. Chang, Experimental study of mass transfer around a turbulence promoter by the limiting current method. In/. J. HcJar A4a.v.~Tru~~sfc~r 26, 1007~1016 (1983). 9. A. D. Gosman. W. M. Pun, A. K. Runchal, D. B. Spalding and M. Wolfshein, 3feot rtnrl Mt~y Tr(it~,~~~~r in Rwir~,~~/~~~i~?,~ F/oI~:F.Academic Press. New York ( 1969). IO. P. K. Chang, Scpcrr~tion o/‘ Floi<~. KlST Press. Seoul (1978). H~~lrod~~~n~i~.t. PrcnticeI I. V. G. Levich. PII1~.~;r~oc~hemi~crl Hall, Englewood Cliffs, New Jersey (1962).

DE MASSE AUTOUR DE PROM~TEURS PLACES DANS UN TUBE CIRCULAIRE

PERTE DE PRESSION ET TRANSFERT PERFORES

DE TURBULENCE

R&sumiL-Les coefficients de frottement et de transfert de masse dans un tube ayant des ailettes de e/D = 0. I: 0,2; 0,24 et 0.3 sont etudies expirimentalement et theoriquement dans un large domaine de nombre de Reynolds jusqu’a 8000. Comme p&u le coefficient de frottement decroit liniairement quand le nombre de Reynolds augmente jusqu’a Rz = 200 et reste a peu pres constant pour Re > 500 dans tous les cas de e/D. La distribution de pression en&e Ies ailettes est mesurCe en detail. Les ail&es perforbes sont efficaces i la fois pour la reduction de la perte de pression et pour l’accroissement de transfert de masse. DRUCKABFALL

IJND STOFFUBERGANG TURBULENZERZEUGERN

IN DER N.&HE VON PERFORIERTEN IN EINEM KREISROHR

Zusammenfassung-Die Reibungsbeiwerte und die Stoffiibergangskoeffrzienten in einem Rohr mit Rippen (cj,!D = 0, I ; 0,2 : 0.24; O,3) werden in einem weiten Bereich der Reynolds-Zahi (bis zu 8000) experimentell und theoretisch untersucht. Wie erwartet, nehmen fur alle Werte von e/n die Reibungsbeiwerte bis zu Re = 200 linear mit dcr Reynolds-Zahl ab und bleiben fur Rr 1 500 annahernd konstant. Die Druckverteilung zwischen den Rippen wird ausfiihrlich gemessen. Die perforierten Rippen erweisen sich sowohl bei der Verminderung des Druckverlustes als such bei der Intensivierung des Stoffiibergangs als wirkungsvoll.

IIEPEIIAJI JIABJIEHHR H MACCOIIEPEHOC BOKPYI- IIEP~OPRPOBAHHbIX TYPBYJIH3ATOPDB, HAXOflCrII&IXC~ B TPYGE KPYIJIOI-0 CEHEHI45I ~Ta~-3KC~epnMe~T~bHO N Ti2O~WleCKH U~AeAO~~Cb KO~H~eHT~ TpeHUX H MacCOIlepe~oca~~py6ecpe6pabsi,y KOTOP~IX e~D=O,1;0,2;424e0,3,~ ~~po~o~A~ana3oHen3MeHeH~~~ncna PeiiffoJIbLWi B~IJIOT~AO 3Ha~etiHir 800O.Kax B OxcHAaJIOCb,K0$J&IuHeHT TpeHHs31HHe&f0 yMeHbIuaJIcR AJIX 3HWIeHHfi %iCAa P&HOJlbACa BMOTb A0 Re= 200 H OCTTBBWlHCb IIOYTH IloCTORHHbIMH IIpSS Re > 500 noun BGZX 3Hawxd e/D. &xanbHo npoMepeH0 pacnpeAenenHe Aaeneiilii~ertny pe6pabw. ~oxa3ario,wo nep@opHpoaarisbre pe6pa~oryT~~~~~~0~cnOAb30naTbcnA.nxyMeHbuxeH~~nOTepb AaeneHARHHHTe~cH~HKaUHB Tennonepexioca.